Explanation:
16. The first attachment shows a table of the given values and the function evaluated at those points.
Maximum: 11 at (4, -3)Minimum: -8 at (-4, 0)
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17. The cost function for this problem is an expression of the total cost as a function of number of days open:
c(x, y) = 40x + 50y
The system of inequalities expresses the constraints on delivery of glass and aluminum in terms of the number of days open:
140x + 100y ≥ 154060x + 180y ≥ 1440
To minimize costs, Center 1 should be open 6 15/16 days; Center 2 should be open 5 11/16 days. The cost function is minimized when it goes through the vertex of the feasible region that puts it closest to the origin.
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18. (a) Jane can use the revenue function ...
r(x, y) = 120x +70y
(b) The constraints on hours and numbers of visits are ...
x + (1/2)y ≤ 8 . . . . . . hours in the dayy ≤ 7 . . . . . . . . . . . . . constraint on follow-up visits
(c) For the given vertices, Jane's best choice is (4, 7), which will produce $970 in revenue for the office.
As is sometimes the case, the integer vertex closest to the corner of the feasible region is not the one that maximizes revenue. Jane's best choice is not on the problem's list. It is (5, 6), which will produce $1020 in revenue.
See the second attachment for the graph related to the problem.
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Apology
The graphs are out of order because my first attempt at 17 had an error. The corrected graph was added as the last attachment.
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Steps
In all of these linear programming problems, the "objective function" is the function of the problem variables that you want to maximize or minimize. In order to write it, you need to understand what the problem variables are and how they relate to the objective. In each of these problems, you are told what x and y stand for and their relation to the objective.
When considering the constraints, you must consider how the problem variables relate to any limits imposed. As in problem 17, sometimes the limits are minima (must deliver at least ...). In problem 18, the limits are maxima (8 hours in a day; no more than 7 follow-ups).
So, first read and understand the problem statement and the relationships it is telling you. Then, do what the problem asks you to do. Sometimes that will involve finding a solution; sometimes not.
Often, you can use logic to help you understand whether your solution is reasonable. In the doctor problem (18), the doctor makes more money per hour doing follow-ups, so would probably want to maximize those (4, 7). However, doing that leaves a half-hour with zero revenue. That last hour is better spent seeing a new patient ($120) than seeing only one follow-up patient ($70).
